Beth number

To define the beth numbers, start by letting

${\displaystyle \beth _{0}=\aleph _{0}}$

be the cardinality of any countably infinite set; for concreteness, take the set ${\displaystyle \mathbb {N} }$ of natural numbers to be a typical case. Denote by P(A) the power set of A; i.e., the set of all subsets of A. Then, define

${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }},}$

which is the cardinality of the power set of A if ${\displaystyle \beth _{\alpha }}$ is the cardinality of A.

Given this definition,

${\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots }$

are respectively the cardinalities of

${\displaystyle \mathbb {N} ,\ P(\mathbb {N} ),\ P(P(\mathbb {N} )),\ P(P(P(\mathbb {N} ))),\ \dots .}$

so that the second beth number ${\displaystyle \beth _{1}}$ is equal to ${\displaystyle {\mathfrak {c}}}$, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number ${\displaystyle \beth _{2}}$ is the cardinality of the power set of the continuum.

Because of Cantor's theorem, each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals, λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:

${\displaystyle \beth _{\lambda }=\sup\{\beth _{\alpha }:\alpha <\lambda \}.}$

One can also show that the von Neumann universes ${\displaystyle V_{\omega +\alpha }}$ have cardinality ${\displaystyle \beth _{\alpha }}$.

Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between ${\displaystyle \aleph _{0}}$ and ${\displaystyle \aleph _{1}}$, it follows that

${\displaystyle \beth _{1}\geq \aleph _{1}.}$

Repeating this argument (see transfinite induction) yields ${\displaystyle \beth _{\alpha }\geq \aleph _{\alpha }}$ for all ordinals ${\displaystyle \alpha }$.

The continuum hypothesis is equivalent to

${\displaystyle \beth _{1}=\aleph _{1}.}$

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., ${\displaystyle \beth _{\alpha }=\aleph _{\alpha }}$ for all ordinals ${\displaystyle \alpha }$.

The more general symbol ${\displaystyle \beth _{\alpha }(\kappa )}$, for ordinals α and cardinals κ, is occasionally used. It is defined by:

${\displaystyle \beth _{0}(\kappa )=\kappa ,}$

${\displaystyle \beth _{\alpha +1}(\kappa )=2^{\beth _{\alpha }(\kappa )},}$

${\displaystyle \beth _{\lambda }(\kappa )=\sup\{\beth _{\alpha }(\kappa ):\alpha <\lambda \}}$ if λ is a limit ordinal.

So

${\displaystyle \beth _{\alpha }=\beth _{\alpha }(\aleph _{0}).}$

In ZF, for any cardinals κ and μ, there is an ordinal α such that:

${\displaystyle \kappa \leq \beth _{\alpha }(\mu ).}$

And in ZF, for any cardinal κ and ordinals α and β:

${\displaystyle \beth _{\beta }(\beth _{\alpha }(\kappa ))=\beth _{\alpha +\beta }(\kappa ).}$

Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

${\displaystyle \beth _{\beta }(\kappa )=\beth _{\beta }(\mu )}$

holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).

This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.

• T. E. Forster, Set Theory with a Universal Set: Exploring an Untyped Universe, Oxford University Press, 1995 — Beth number is defined on page 5.
• Bell, John Lane; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974 ed.). Dover Publications. ISBN 0-486-44979-3. See pages 6 and 204–205 for beth numbers.
• Roitman, Judith (2011). Introduction to Modern Set Theory. Virginia Commonwealth University. ISBN 978-0-9824062-4-3. See page 109 for beth numbers.